Vertex-degree function index for concave functions of graphs with a given clique number.

For any connected graph G = (V , E) and any function f on the positive integers set Z + , vertex-degree function index H f (G) is defined as the sum of f (d G (v)) over v ∈ V , where d G (v) is the degree of v in G. For any n , k ∈ Z + with n ≥ k , connected graphs with n vertices and clique number k form the set W n , k . In this paper, for any strictly concave and increasing function f on Z + , we determine the maximal and minimal values of H f (G) over G ∈ W n , k , and characterize the corresponding graphs G ∈ W n , k with the extremal values. We also get the maximum vertex-degree function index H f (G) , where f(x) is a strictly concave and decreasing function for x ≥ 1 . [ABSTRACT FROM AUTHOR]